Discrete Dynamic Systems

What is a Dynamical System?

Definition of Dynamical System  A dynamical system is characterized by a ‘rule’ (or ‘rules’) that determines how points in the state space of the system change over time  The particular form of a rule depends on the type of dynamical system being studied, and the type of dynamical system depends on the space– time structure of the system.

Two Geometrical Cases of Dynamical Systems

Sequences, Bifurcations, and Period Doubling

Sequences in the Logistic Family RnxRnxRnx

Period Doubling in the Logistic Family  Plots from IterateMapApp.java  Red – R = 0.4  Green – R = 0.6  Blue – R = 0.8  Note how blue seems to oscillate between two equilibria Figure 1: IterateMapApp.java

Bifurcations in the Logistic Family Figure 2: BifurcateApp.java

Chaos

What is Chaos?  “Chaos: When the present determines the future, but the approximate present does not approximately determine the future,” Edward Lorentz  Chaos occurs when there is no repeating sequence in later time

How Do We Quantify Chaos?  Lyapunov Exponents Figure 3: BifurcateAppDDP3.java

Weakly, Strongly, and Non Chaotic  The continuous and discrete equations describe the same system. However… 1) In continuous we only need to worry about the roughness in the inputs/ initial conditions. But in discrete we also have to worry about the roughness in our time step.